Question

The following table shows the marks scored by 140 students in an examination of a certain paper:

Marks:	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50
Number of students:	20	24	40	36	20

Calculate the average marks by using all the three methods: direct method, assumed mean deviation and shortcut method.

Answer 1

From Direct method:

Class interval	Mid value (xi)	fi	fixi
0 - 10	5	20	100
10 - 20	15	24	360
20 - 30	25	40	1000
40 - 50	45	20	900
		N = 140	\(\sum\)fixi = 3620

Mean = \(\frac{\sum f_ix_i}{N}\)

\(=\frac{3620}{140}=25.857\)

Assumed mean method: 

let assumed mean (A) = 25

Mean = A + \(\frac{\sum f_iu_i}{N}\)

Class interval	Mid value (xi)	ui = xi - A	fi	fiui
0 - 10	5	-20	20	-400
10 - 20	15	-10	24	-240
20 - 30	25	0	40	0
30 - 40	35	10	36	360
40 - 50	45	20	20	400
			N = 140	\(\sum\)fiui = 120

Mean = A + \(\frac{\sum f_ix_i}{N}\)

\(=25+\frac{120}{140}=25+0.857\)

\(=25.857\)

Step deviation method: Let the assumed mean (A) = 25

Class interval	Mid value (xi)	di = xi - A = xi - 25	ui = \(\frac{xi-25}{10}\)	Frequency (fi)	fiui
0 - 10	5	-20	-2	20	-40
10 - 20	15	-10	-1	24	-24
20 - 30	25	0	0	40	0
30 - 40	35	10	1	36	36
40 - 50	45	20	2	20	40
				N = 140	\(\sum\)fiui = 12

Mean = A + \(\frac{\sum f_iu_i}{N}\times h\)

\(=25+0.857=25.857\)

\(=25+\frac{12}{140}\times10\)

\(=25+0.857=25.857\)